Calculus & Analysis on

Earliest Known Uses of Some of the Words of Mathematics

Earliest Uses of Various Mathematical Symbols

The following is a list of entries on the Words page in the general area of DIFFERENTIAL CALCULUS and ANALYSIS. Mathematical analysis is a subject of enormous scope. In the course of the 19^th century COMPLEX ANALYSIS became established as a major field and the 20^th century saw the rise of FUNCTIONAL ANALYSIS.

Sy next to an item is a link to related material on the Symbols pages, principally Earliest Uses of Symbols of Calculus, Earliest Uses of Function Symbols and Earliest Uses of Symbols of Set Theory and Logic.

You may also want to consult the indexes for entries on Set Theory and Logic, Probability & Statistics and Vector Analysis. For a general perspective on word-formation in mathematics see my Mathematical Words: Origins and Sources.

A – B A Abelian integral Absolute continuity Absolute convergence Absolute differential calculus Analysis Analytic function Anti-derivative Anti-differential Axiom Axiom of choice B Baire classes Banach space Banach-Steinhaus theorem Banach-Taski paradox Bernoulli’s equation Bessel function Sy Beta & gamma functions Sy Bijection Brachistochrone	C C Calculus Calculus of variations Cantor set Cartesian product Catastrophe theory Catenary Cauchy convergence Cauchy-Schwarz inequality Cauchy-Riemann equations Cesáro mean Chain rule Chaos Circle of convergence Circular functions Clairaut equation Clairaut’s theorem Closed set Compact Complementary function Completeness Complex analysis Complex number Sy Sy Sy Conditionally convergent Conformal mapping Conjugate Constant of integration Continuous Convergent Convolution Cotangent space Covariant differentiation Covering Curl	D – E D Daniell integral Definite integral Sy Del Sy Dense Dependent variable Derivative Sy Differential Differential calculus Differential equation Differential geometry Differential topology Differentiation Dirac δ function Directional derivative Dirichlet integral Dirichlet’s principle Distribution Divergence Sy Divergence theorem Divergent Domain Double integration Dummy variable E Element Sy Elliptic function Empty set Sy Euler’s equation Eulerian integral Euler-Maclaurin summation Exact differential Explicit function Exponential function Extremal Extremum
F – G F Fast Fourier transform Fejér kernel First derivative Fluxion & fluent Fourier series Fourier’s theorem Fourier transform Fractal Fréchet differential Fubini’s theorem Function Sy Functional Functional analysis Functional calculus Fundamental theorem of calculus G Gamma function Gauss’s theorem Gaussian curvature General integral General solution Generalized function Geodesic Geometric series Gibbs phenomenon Gram-Schmidt orthogonalization Gradient Green’s theorem Gregory’s series Gregory-Newton formula	H – J H Haar measure Hahn-Banach theorem Hamiltonian Hard and Soft Analysis Hardy classes and spaces Hardy-Littlewood theorem Harmonic analysis Harmonic function Hausdforff measure Hausdorff paradox Hausdorff space Heine-Borel theorem Hermite polynomial Hessian Hilbert space Hölder mean Holomorphic function Homogeneous differential equation Hyperset I Imaginary Implicit differentiation Implicit function Improper integral Indefinite integral Infinitesimal Infinity Sy Integral Sy Integral calculus Integral equation Inverse J Jacobian Jordan curve Julia set	K - N K Kernel Kolmogorov extension theorem Kuhn-Tucker theorem L Lagrange multiplier Laplace transform Laplacian Laurent expansion Lebesgue integral L’Hospital’s rule Limit Sy Limit point Linear differential equation Logarithm Sy Lower semicontiuity Lorenz attractor M Maclaurin’s series Manifold Mapping Measurable function Measure Method of exhaustion Metric space Modulus Monotonic Morse theory  N Nabla Sy Neighborhood Newton’s method Non-standard analysis Norm Numerical differentiation Numerical integration
O – P O O and o notation Sy One-to one correspondence Onto Open set Ordinary differential equation Oscillating series P Paradox Partial derivative Sy Partial solution Pfaffian Point-set Point of accumulation Pole Potential function Power series Primitive function Projection	Q – R Q Quadrature Quasi-periodic function R Radius of curvature Radon-Nikodym theorem Range (of a function) Rational function Real analysis Real number Riccati equation Riemann integral Riemann zeta function Riesz-Fischer theorem Rolle’s theorem Runge-Kutta method	S S Schlicht Schwarz inequality Schwarz’ theorem Sequence Series Set and Set theory Simpson’s rule Single-values function Singular integral Singular point Slope field Space Spectrum Stieltjes integral Stirling’s formula Stokes’ theorem Strange attractor Subset Summable Surface integral Surjection
T T Tauberian theorems Taylor’s formula, theorem series Tensor Theorem Transfinite	U – V U Uniform continuity Uniform convergence Univalent V Variable Sy	W – Z W Wallis’ formula Weierstrass approximation Wiener-Hopf Witch of Agnesi Wronskian Y Young’s criterion Young’s theorem Z Zeno’s paradoxes Zorn’s lemma

History

Although the mathematicians of antiquity calculated areas by the method of METHOD OF EXHAUSTION, it is true nonetheless that “analysis as an independent subject was created in the 17^th century during the scientific revolution.” (Jahnke). For three hundred year it has been one of the most important branches of higher mathematics and has attracted the attention of many of the greatest mathematicians including Newton and Leibniz in the 17^th century, Euler and Lagrange in the 18^th century and Cauchy and Weierstrass in the 19^th century. The entry on DIFFERENTIAL CALCULUS shows how the scope of the subject broadened and how the original differential calculus interacted with other branches of mathematics, such as geometry and topology.

Analysis is an important subject and many accounts of its history are available. Its history is a prominent topic in general works on the history of mathematics, such as Katz and Kline, and there are many more specialised works, such as Jahnke and Hawkins and Grabiner. The chapters of Jahnke contain extensive bibliographies.

References

• Morris Kline Mathematical Thought from Ancient to Modern Times. Oxford University Press, 1972. Reprinted as 3 volumes. Amazon, Amazon, Amazon.
• Victor J. Katz, A History of Mathematics: An Introduction. Addison Wesley, 1998. Amazon.
• Hans Niels Jahnke (ed.) A History of Analysis. American Mathematical Society 2003. Amazon
• Thomas Hawkins Lebesgue’s Theory of Integration: Its Origins and Development, AMS Chelsea Publishing Series, 1979/2001 Amazon.
• Judith V. Grabiner The Origins of Cauchy's Rigorous Calculus, Dover, 1981/2005. Amazon

On the web

A lot of material is available but the following make a useful start.

• James Stewart : History of Mathematics
• David E. Joyce : History of Analysis
• MacTutor: A history of the calculus
• MacTutor: Analysis topics
• Encyclopaedia of Mathematics : Mathematical analysis
• Encyclopaedia of Mathematics : Differential calculus
• Encyclopaedia of Mathematics : Infinitesimal calculus
• Encyclopaedia of Mathematics: Theory of functions of a complex variable
• Encyclopaedia of Mathematics: Functional analysis

John Aldrich, University of Southampton, Southampton, UK. (home). Most recent changes October 2009.